Coupler-curve synthesis of four-bar linkages via a novel formulation

• We show that, in four-bar linkage coupler curve synthesis, the system of coefficient equations in the nine linkage parameters is determined.
• A system of seven equations with seven unknowns is formulated for the coupler-curve synthesis
• A new method combining a numerical method with graphics tools allows us to reduce the complexity of solving a system of equations.
• The new method is able to synthesis simultaneously the three cognate linkages

The coupler-curve synthesis of four-bar linkages is a fundamental problem in kinematics. According to the Roberts–Chebyshev theorem, three cognate linkages can generate the same coupler curve. While the problem of linkage synthesis for coupler-curve generation is determined, it has been regarded as overdetermined, given that the number of coefficients in an algebraic coupler-curve equation exceeds that of linkage parameters available. In this paper, we develop a new formulation of the synthesis problem, whereby the linkage parameters are determined “exactly”, within unavoidable roundoff error. A system of coupler-curve coefficient equations is derived, with as many equations as unknowns. The system is thus determined, which leads to exact solutions for the linkage parameters. A method of linkage synthesis from a known coupler-curve equation is further developed to find the three cognate mechanisms predicted by the Roberts–Chebyshev theorem. An example is included to demonstrate the method.